Explanation Proof: The frustum of the cone will be obtained when a straight line passing through points ( R 2 , 0 ) and ( R 1 , h ) is revolved about the y-axis in the interval 0 ≤ y ≤ h . The equation of the line will be, y − y 1 = ( y 2 − y 1 x 2 − x 1 ) ( x 2 − x 1 ) y − 0 = h − 0 R 1 − R 2 ( x − R 2 ) y = h R 1 − R 2 ( x − R 2 ) Solve the above equation for x, x − R 2 = y R 1 − R 2 h x = R 2 + ( R 1 − R 2 ) h y The volume generated will be calculated using Disk Method, If D is the region bounded by the curve x = f ( y ) , the y-axis and the horizontal lines y = c and y = d. Then If D is revolved about the y-axis, it generates a solid with volume V = ∫ c d π [ f ( y ) ] 2 d y Substitute, the values in the above expression, f ( y ) = R 2 + ( R 1 − R 2 ) h y , c = 0 And d = h The volume generated will be, V = ∫ 0 h π [ R 2 + ( R 1 − R 2 ) h y ] 2 d y = π ∫ 0 h [ R 2 2 + 2 R 2 + ( R 1 − R 2 ) h y + R 2 + ( R 1 − R 2 ) h y ] d y = π [ { R 2 2 y + 2 R 2 + ( R 1 − R 2 ) h × h 2 2 + ( R 1 − R 2 ) h 2 × h 3 3 } − R 2 2 0 + 2 R 2 + ( R 1 − R 2 ) h × 0 + ( R 1 − R 2 ) h 2 × 0 ] Simplify further, V = π [

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.2, Problem 60PS

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To show:The volume of thefrustum is V=h3(A1+A1A2+A2) .

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Chapter 6.2, Problem 59PS

Chapter 6.3, Problem 1PS

Chapter 6 Solutions

Calculus

Ch. 6.1 - Prob. 1PS Ch. 6.1 - Prob. 2PS Ch. 6.1 - Prob. 3PS Ch. 6.1 - Prob. 4PS Ch. 6.1 - Prob. 5PS Ch. 6.1 - Prob. 6PS Ch. 6.1 - Prob. 7PS Ch. 6.1 - Prob. 8PS Ch. 6.1 - Prob. 9PS Ch. 6.1 - Prob. 10PS

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The volume of thefrustum is V = h 3 ( A 1 + A 1 A 2 + A 2 ) .

Solution Summary: The author explains how the frustum of the cone will be obtained when a straight line passing through points (R_2,0) is revolved about y-

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To show:The volume of thefrustum is V=h3(A1+A1A2+A2) .

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Chapter 6 Solutions